The coupled ocean-atmosphere system has instabilities that span time scales from a few minutes (e.g. cumulus convection) to years (e.g. El Ni$\tilde{n}$o). Fast time scales have stronger growth rates and within linear approximations used in data assimilation, they do not saturate and may distort the slower longer time-scale solution. Therefore, it is not clear whether a data assimilation focused on long-term variability should include smaller time scales.

To study this problem, we perform sequential and variational data assimilation experiments with 3 coupled Lorenz (1963) models of different time scales, simulating a coupled ocean-atmosphere model. We aim to better understand the abilities of different data assimilation methods for coupled models and aid in the development of data assimilation systems for larger coupled ocean-atmosphere models such as a general circulation models.

The dissertation provides an overview of the sequential and variational data assimilation methods, which includes Ensemble Kalman Filter (EnKF)-based methods, a fully coupled 4-dimensional variational data assimilation (4D-Var), and an ECCO-like 4D-Var, which uses the initial ocean state and surface fluxes as control variables. Assuming a perfect model and observing all model variables, Ensemble Kalman Filter (ENKF)-based algorithms without a quasi-outer loop or model localization experienced filter divergence for long assimilation windows, but were stable for shorter windows. The EnKF analyses depend on the covariance inflation and number of ensemble members. We found that short assimilation windows require a smaller inflation than long assimilation windows. The fully coupled 4D-Var analyses provide a good estimate of the model state and depend on the amplitude of the background error covariance. When comparing the EnKF analyses with the 4D-Var analyses, we found that the filters with a quasi-outer loop and model localization are more accurate than the fully coupled 4D-Var analyses for short windows, but the fully coupled 4D-Var method outperforms the EnKFs for long windows. The ECCO-like 4D-Var improves the 4D-Var analyses which uses only the initial ocean state as control variables, but the fully coupled 4D-Var outperforms the ECCO-like 4D-Var and 4D-Var analyses. The data assimilation experiments offer insight on developing and advancing sequential and variational data assimilation systems for coupled models.